Timing references in electronics have been realised mechanically for a long time. Quartz crystal resonators in an oscillator package are present in many applications. The high quality factor and low temperature drift of quartz crystal resonators means that they have high stability and are therefore chosen as a timing reference in electronics.
MEMS resonators, on the other hand are devices formed from miniaturised components operatively arranged on a substrate. MEMS resonators are typically constructed through the use of lithographic and other micro-fabrication techniques to produce, for example, sensors and actuators. Micromechanical resonators are generally formed on a silicon substrate of the type used for integrated circuits, and may be manufactured using CMOS technology.
Recently MEMS resonators have been developed with a view to replacing quartz in the timing market. These resonators have high Q factors (which is a measure for frequency selectivity) combined with extreme form factors. Typical dimensions of conventional quartz oscillator packages are in millimeters, whereas MEMS oscillators comprising a resonator and driving electronics can be fabricated in thin-film technology leading to a thickness of less than 100 μm. However the beneficial form factor of a MEMS resonator alone is not enough to make it a candidate for most timing reference applications. This is because the resonance frequency of a MEMS resonator is temperature dependent. This means that the resonance frequency will not be constant over an operating temperature range.
Although there are many factors that affect the temperature dependency of the frequency of a resonator, the temperature dependent modulus of elasticity (or Young's modulus) of silicon, the material most often used to form the resonator, largely determines the temperature coefficient of the resonator. When a resonator is formed from silicon, the nominal value of the Young's modulus in the <100> direction, together with the negative temperature dependency of silicon is well known. This means that the frequency of a resonator vibrating solely in the <100> direction can be predicted accurately for a given geometry at room temperature. Because of the negative temperature coefficient (TC) of the Young's modulus, the frequency of the resonator also has a negative TC.
In order to overcome the problem of the temperature dependence of the resonant frequency of a MEMS resonator having a negative temperature coefficient, it is known to use passive temperature compensation techniques. One such passive technique is to add a material of positive temperature coefficient to such a resonator. By adding such a material to the resonator, the overall temperature coefficient may become less negative, and within certain tolerances may be reduced to zero. Under such circumstances the frequency of the resonator would become temperature independent.
It is known that silicon dioxide possesses a positive temperature coefficient and it is known to coat a resonator formed from silicon with a silicon dioxide skin in order to compensate for the dependence of the resonant frequency.
One known method comprises the steps of applying a skin of silicon dioxide around a suspended silicon resonator. This method is referred to as global oxidation since all the silicon over the entire surface of the resonator (top area, bottom area, and sidewalls) will be transferred to silicon dioxide at substantially the same rate.
Another known method is known as local oxidation in which only a part of the silicon resonator is either transferred to, or replaced by silicon dioxide.
It is known that resonators formed solely from silicon exhibit a negative temperature drift of 30 ppm/K on their resonance frequency. This means that over a range of 100° C., the frequency will change by 3000 ppm. This value of −30 ppm/K is referred to as the linear temperature coefficient (TC) of the resonator.
After a resonator formed from silicon has been coated with a silicon dioxide layer, the linear temperature coefficient has a close to linear relationship with the thickness of the grown oxide layer.
For any such resonator, there will exist an oxide layer thickness that results in the linear TC being zero. This results in the frequency of the resonator no longer varying substantially with temperature and thus being almost temperature independent.
However, the frequency of the resonator will depend on other factors as well as on the ambient temperature. This means that if the variance of the frequency with respect to temperature is reduced to a small value such that the TC is close to zero, it is still not possible to produce a wafer comprising resonators having exactly the same resonance frequency as one another, since the absolute frequency of given resonator will depend, inter alia on the thickness of the oxide layer. For an oxide thickness of 300 nm, the change in frequency relative to the frequency of an unoxidised resonator, ignoring any variation caused by temperature is over 100000 ppm. It can be seen therefore that the effects on frequency caused by a variation in oxide thickness are much larger than the changes in frequency over a 100° temperature range.
Since the frequency is highly dependent on the oxide thickness, it is unsurprising that large frequency spreads exist in MEMS resonators having a thick oxide.
In other words, whilst an oxide layer may reduce the TC of a resonator, the frequency of the resonator will nevertheless vary with variations in the thickness of the layer.
Silicon dioxide is usually applied using a thermal oxidation process. Such a process has a relative error of a few percent. For example, for an intended oxide layer thickness of 300 nm, the layer will in fact have a thickness of between 293 nm and 308 nm meaning that there is a relative error of 5%. This means that the resulting absolute frequency of the resonator may have a spread falling within the range of thousands of parts per million. This range is too large for most applications.
The frequency of a resonator will also depend on lithographic/etch variations known as Δ as both the spring constant and the mass are sensitive to geometrical variations in, respectively, the spring and the mass regions of the resonator.
As is well known in the art, photolithographic techniques may be used to define features having extremely small geometries of the type required in a MEMS resonator. These features may be formed by initially coating a substrate (typically silicon) with a thin layer of a photosensitive material, transferring a pattern onto the photosensitive layer using a known exposure technique and then etching the pattern onto the photosensitive layer to remove the exposed portions, revealing corresponding portions of the underlying substrate. These exposed portions of the substrate may then be processed. This process may be repeated to create further layers as necessary.
A problem associated with this technique is that under or over exposure of the photosensitive material may occur. This may cause incomplete transfer of a pattern, or inaccurate line widths of exposed portions of the structure. Subsequent physical or chemical etching of the exposed regions is also susceptible to variations due to e.g. variations in the compositions of etch chemistries, etch chamber pressures, gas flow rates, etc. These variations can all result in geometric variations.
Here “Δ” is used to denote variations in one dimension from the nominal geometry of a resonator due to process variations during lithography and etch. It is assumed that these variations are “non-local” and are isotropic. Thus “Δ” is the same for all regions in an individual device, or, in other words, if the spring is 10 nm narrower than intended, then the mass is both 10 nm shorter and 10 nm narrower than intended. This assumption is reasonable given that the spread-free designs, based on the same assumption, are demonstrated to reduce frequency spread.
The absolute resonant frequency is very sensitive to variations in the oxide thickness tox, and/or post-etch geometry Δ so even the best controlled processes may result in unacceptable frequency spread.
Another process variation that may affect the frequency of the resonator is the thickness of the substrate forming the resonator. Typically, for a Silicon on Insulator (SOI) substrate formed from silicon, the thickness of the substrate TSOI may vary within a range of approximately 3%.